{
 "metadata": {
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.5-final"
  },
  "orig_nbformat": 2,
  "kernelspec": {
   "name": "python38564bit248ee495c5e64151b0d3dda955bfc768",
   "display_name": "Python 3.8.5 64-bit"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2,
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sym\n",
    "import numpy as np\n",
    "from matplotlib import pyplot as plt\n",
    "sym.init_printing()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "x,h,t = sym.symbols('x,h,t')\n",
    "v0 = sym.symbols('v_0') # initial velocity\n",
    "vc = sym.symbols('v_c') # climb velocity\n",
    "dv = sym.symbols('dv') # climb velocity adjustment\n",
    "dx = sym.symbols('dx') # x distance to point\n",
    "dh = sym.symbols('dh') # altitude difference\n",
    "v1 = sym.symbols('v_1') # final velocity\n",
    "x0 = sym.symbols('x_0') # Distance to transition from v0 to vc\n",
    "x1 = sym.symbols('x_1') # Distance to transition from v0 to vc\n",
    "x2 = sym.symbols('x_2') # Distance transitioned during climb accel\n",
    "x3 = sym.symbols('x_3') # Distance transitioned during steady climb\n",
    "x4 = sym.symbols('x_4') # Distance at end of climb\n",
    "x5 = sym.symbols('x_5') # Distance to transition from vc to v1\n",
    "h0 = sym.symbols('h_0') # Initial altitude\n",
    "h1 = sym.symbols('h_1') # Altitude at x1\n",
    "h2 = sym.symbols('h_2') # Altitude at x2\n",
    "h3 = sym.symbols('h_3') # Altitude at x3\n",
    "h4 = sym.symbols('h_4') # Altitude at x4\n",
    "h5 = sym.symbols('h_5') # Altitude at x5\n",
    "t0 = sym.symbols('t_0') # time at x0\n",
    "t1 = sym.symbols('t_1') # time at x1\n",
    "t2 = sym.symbols('t_2') # time at x2\n",
    "t3 = sym.symbols('t_3') # time at x3\n",
    "t4 = sym.symbols('t_4') # time at x4\n",
    "t5 = sym.symbols('t_5') # time at x5\n",
    "ax = sym.symbols('a_x') # horizontal acceleration\n",
    "az = sym.symbols('a_z') # vertical acceleration\n",
    "dax = sym.symbols('da_x') # horizontal acceleration\n",
    "daz = sym.symbols('da_z') # vertical acceleration\n",
    "hdot = sym.symbols('\\dot{h}') # Vertical speed\n",
    "hdot0 = sym.symbols('\\dot{h}_0') # Initial Vertical speed"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Ground values for free variables\n",
    "_v0 = 1\n",
    "_vc = 0.8\n",
    "_v1 = 1\n",
    "_ax = 1.5\n",
    "_az = 1\n",
    "_hdot = 2.5\n",
    "_hdot0 = 0\n",
    "_h0  = 0\n",
    "_x0 = 0\n",
    "_t0 = 0\n",
    "_dax = -1\n",
    "_daz = -1\n",
    "_dx = dx.subs(_x0+dx,8.6423)\n",
    "_dh = dh.subs(_h0+dh,10)\n",
    "sub = {v0:_v0,vc:_vc,ax:_ax,az:_az,hdot:_hdot,hdot0:_hdot0,h0:_h0,dx:_dx,dh:_dh,x0:_x0,t0:_t0,dax:_dax,daz:_daz,h4:_h0+_dh,h5:_h0+_dh,v1:_v1}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "          -v₀ + v_c\nt₁ = t₀ + ─────────\n             daₓ   ",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle t_{1} = t_{0} + \\frac{- v_{0} + v_{c}}{da_{x}}$"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "                                          2\n          v₀⋅(-v₀ + v_c)   0.5⋅(-v₀ + v_c) \nx₁ = x₀ + ────────────── + ────────────────\n               daₓ               daₓ       ",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle x_{1} = x_{0} + \\frac{v_{0} \\left(- v_{0} + v_{c}\\right)}{da_{x}} + \\frac{0.5 \\left(- v_{0} + v_{c}\\right)^{2}}{da_{x}}$"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "h₁ = h₀",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle h_{1} = h_{0}$"
     },
     "metadata": {}
    },
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "(0.2, 0.18, 0)",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle \\left( 0.2, \\  0.18, \\  0\\right)$"
     },
     "metadata": {},
     "execution_count": 34
    }
   ],
   "source": [
    "# Using this equation: v = u + a*t\n",
    "# Time to reach x1 from x0 with acceleration ax: (ax is deceleration if vc < v0)\n",
    "dt_01 = (vc-v0)/(dax) \n",
    "\n",
    "# Using equation: s = ut + 1/2*a*t^2\n",
    "# Distance traveled from x0 to x1\n",
    "dx_01 = v0*dt_01 + 1/2*(dax)*dt_01**2 \n",
    "\n",
    "exp6 = sym.Eq(x1,x0 + dx_01)\n",
    "exph01 = sym.Eq(h1,h0)\n",
    "exp7 = sym.Eq(t1,t0 + dt_01)\n",
    "\n",
    "display(exp7,exp6,exph01)\n",
    "\n",
    "_t1 = exp7.args[1].subs(sub); sub[t1] = _t1\n",
    "_x1 = exp6.args[1].subs(sub); sub[x1] = _x1\n",
    "_h1 = exph01.args[1].subs(sub); sub[h1] = _h1\n",
    "_t1,_x1,_h1\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "                                        2     \nh = \\dot{h}₀⋅(t - t₁) + 0.5⋅a_z⋅(t - t₁)  + h₀",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle h = \\dot{h}_0 \\left(t - t_{1}\\right) + 0.5 a_{z} \\left(t - t_{1}\\right)^{2} + h_{0}$"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "x = v_c⋅(t - t₁) + x₁",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle x = v_{c} \\left(t - t_{1}\\right) + x_{1}$"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "     \\dot{h}     \nt₂ = ─────── + t₁\n       a_z       ",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle t_{2} = \\frac{\\dot{h}}{a_{z}} + t_{1}$"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "                                             2     \nh₂ = \\dot{h}₀⋅(-t₁ + t₂) + 0.5⋅a_z⋅(-t₁ + t₂)  + h₀",
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYIAAAAaCAYAAAC6uFsTAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAKE0lEQVR4Ae2d25XUOBCGezgTwCxEsJDBsEQAZMAlAiADODzB2xw2AyCChc0AiACWDCADhslg9v/UkrHdsi3JUnebVp2j0cW6lH6VSiVZ7Tm6vLxcVaoIVAQqAhWBaQRevnx5olzPbc7r1n+k9Ivp0vub43h/WUvjTAPyTSU/yn+SVkMtVRGoCFQEBhF41dYtCr9Wzv/kbgyWWMCDKwvgMZhFDQqrNav0X8GFasaKQEWgIhCOwGPpmTut7K8Uvq6001ba4oK/20LA9uwPDcrNxY1EZbgiUBFYAgKcNHxZAqMxPB7VdwQxcNW8FYGKQEXgFwIyOtkR3JNfj4Z+wXJYIQ2+e1l0WB2f2duK20wAA4pXjANA8mSJwU15OQ66J7f4E4jf6mjIM67FkiQET1X5os8Fi4EzXTFnquC3aLKKoNMHpZ3I7dRAsNhW2eyMTHAkSDbtGLMbuKnwom8MgYw5GlJHeMn6Xo6XrOeKL26bs80+qC2sgFvyn8mvlICAsGMh+C7/34Ti2YqofSYz9EMOuedWyHcSpkj5fioPc+erzUsY2plyEE9VNtdjkPxXGI7Kpp6z0CMn92lEPovuhfwguaFMLlKbWXR35x2BKuUa1Bf5i716WboPFvhP8otsB239jMNdhbcuWKECmoNP1UE/b8u/CG03Zz7b/pl8sxjJZ1IFY6/8XFWGUAyMFfVQ3676A/9VNiPHUZhtkMbQK5tKZ6y5Mto2AtGXz3Y17jBv+U3W3f3fEbCynVHxgql0H7AgEYRSxNU0tqfFFgHV7dp4M6MTro45fIIjeCYbHql9UbnHapdjnGZHojBWHXH4uis3RV+V31iFUxm39LzK5hrokrLJAuEMhmZYJQdJMqxyjtc5cxE+Zum9K64nliGiH13a0vwt9eGB2pk7aGPQooDcUcNYvjnPEGTcHJrNp8URPOfwktoXFLgP589KvzOTpzm4zilbZXONXjHZlFxwPf2o72YMWqr8Nk2KFxYTKFl3NwuBKgE8zmwvqHGhVLQPFvA5FnAIrAxq8oCGNJApTy4+wfNBJp5iqoH/c08BJ/9ucnmy7F9Slc3OmCxdNjudCYjM1nvHrUYMeBIoXjZdlWOlooEnSiut/NRMFirdB6zI7EraTmK2lmDOGeSp0nh5/1n+3/L3ggrxCZ7IWcldVgc/9QOcp4g5MEmqyxwx2Yy8bH6ttI2dhtIYV86VWWh+yHEExf3zXONbZXN9xJhzDm1dNiUTKTRb75mFQMIIeJwxQc2tCaXfIi6HkM0m1cfZK0zHUNA57Jb6wK2q7O8HxDsCx/eRWIQ5lkAx7h0V4pMXrrEyMRcbp+RRyn1yuwTmxBSR551wMfXIR9l/Y/wsVqa8wvSPhZ3bRMaokv9Bccb6jZyPD1M24k+VzfxzaBeyGTHk5iUxMjhbd5uFQBW5ichX9PrWPw01pOfEn9sEBB8K+vqeyia9UFk3Mfk3uA/UJF5Y4CCss2uKt28BmAeeP/TdKQrP49lJLAAb1iS1ij/a5giFHVqRG0u0E0hePi2PsbIBnh0ZC+ShdLZrUw2ov50FW3GOVlnUMRbMFWyLCYsAt0rac4twzqPYKpsCVHQIsrnu6fpvFr3nFgIDngS1r4Ro5KzdqsL7+vW94D6onywCP+Sbbbl8LLMPcp2J3es3USzJC0/6SmWZiJ/kYpTafZVrYw7ezS0W147ysOK7AQ+uX+VQSK6cqw7fWMR67luYQ3ZgXj5Vb4psoBCdQQFvXsrcFxafITLY6CEGQgrRH+TJ3fxC1hizd73KhjDsZQuOVtlcQzWEazHZDBmhzPLbbjKL3ju2NW6AhyDrGQKMhdOmx3r2Xs6lI+ikca7dVmrtMtsIR/VBDN12TNEXORYCN3ndo2BfZS+UOdlSV3mwBnOODDqkZ+CKguboKJiU36foV7Ye+hp9Pq0yg3yKsRTZGFRg7Y6q3Wx9UV1cE6V6+tInl9a23vt5VirPOF2VPzTmrh52ccjXhatEYZ55x9rlyenbtof4nGxqjF89O3jZnARQGYRTNvnttZdF7x2PDDKMs3XtK3fSv/SYCYqqriELdaz8pIUa0wfldQtcMzFt48RP5cYUAM+YxCWIM96V+HMLLGHuuff5LNF2TJ1jfKbIBniexzCQKS84Iwt9cjuCZhz6GWwcHHx8m/IaN+SWvuH6c8jcktLzjTaUhgxylMS4824OHs+pT/4YVdm0n59v46qwm0NLks2xcW6e0TdFfAbFhu5W3lG9d6yKhoSS2xDmJocqIWxeiincv92x0ajyekllyVuCgvugxgEP6k9i4k4JmAyeP+TxKQ9P1ugks8VzpYQV1j+TH4WwTzTIZ6JsgPnY4luq7yhb956o3QaWM0p8Cnde8vreK2Gh9RX853YDCqPgGduV6jDjLN9h8NA+57GZLwPt8LxNVTZ77wcctgKJHWCK3tqVbLbHdSycTe9dUStM7L7g0jgK8x8CohsCcmNiKA3rBUFO3nZSeQZK7kOrbQadPo8Rk5ebVCWIBcYoA+EKH8nHVCWYa9UZxGeEbCA7G8dhrfaKBMUfigFLG/k1ZHFncj2ySSvS5C7l+EVpm7gmyg63IcWf2oi5Zac4cwaZaYwHpWFUEXe7ar5Z5RaBlcK8VKaMmW/EFQ6hKptrXCfnkDAN1Vs7kc2QwbZ5suk9dgQoHayjPnF+/FCg3fEJo9IQZiyqnX1gq8VwTB/MBFNZFL8LUxV1NBOSBA+hsDqT35MnNYkJj3JBUazkR5/fpzYcWW6Sz0jZwIL2WeaRbCVlZ6I/t/xSAYs83z5CqRpSGGsSuXCK26VzbMoLSCcPyNO53J+UMZnWf3gX9VZpLBLX5DCumMCUJc0ZWwquSekoKu+8c3k8fpXN9e80RueQsI3RW7uUTc8QbyShs0J1t5NJv97jH9PEuhcvXlyXe+/KKXxKmovvuy9ef8Jzm0/FL+Um+6A83+Q6Zdv1lAyr3Xu0P7cNW8/TufX4yqvuYNmweWf1p2RffP0rnab+MJeasbEYnYS0q7xVNkf0mcUySG/ZvLNkM3DMmNPNeIeUSc2jdgb1HjuCKLIrKlYQW1gsF4izzNAtrCmw4z8cC7DaG8tP/TBh+VM7AtjGeqW/pd530EZpwjpwFkK2thJkA5mZuxso0pdsoERUZPF7qyJ8wZS5hfXG70bMUVNAVVU2B0DakWwOcNNJ3qb8Duq9zmeoO+wNRASo+wZ7J4fSjzoJex4Rv04BcV+cH/+wsAUpR+VjG761T2+oPbazLDwsWCgIBpRfsO7V8ZH4CZYN2ye28RyTVBICwgKZPJPjPQVhDBN+axJioCirqaPKpkGi+0cYVtlcy4dX70UvBF14DzMmoTpRz/ktRVViiSIg7FBYKLmgxTexmYMrVmVz/pAfomzWhSBRbiQsWOm80MM6rxSBgDDjhTg/sgq2dCOqP/isVTbTReBQZfN/7F82bCPTMREAAAAASUVORK5CYII=\n",
      "text/latex": "$\\displaystyle h_{2} = \\dot{h}_0 \\left(- t_{1} + t_{2}\\right) + 0.5 a_{z} \\left(- t_{1} + t_{2}\\right)^{2} + h_{0}$"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "x₂ = v_c⋅(-t₁ + t₂) + x₁",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle x_{2} = v_{c} \\left(- t_{1} + t_{2}\\right) + x_{1}$"
     },
     "metadata": {}
    },
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "(2.7, 2.18, 3.125)",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle \\left( 2.7, \\  2.18, \\  3.125\\right)$"
     },
     "metadata": {},
     "execution_count": 35
    }
   ],
   "source": [
    "# Expression to compute h during initial vertical acceleration\n",
    "# v = u + at\n",
    "dt_12 = hdot/az\n",
    "# For all t \\in [t1,t2], the following is true for h and x\n",
    "exp4 = sym.Eq(h,h0 + hdot0*(t-t1) + 0.5*az*(t-t1)**2)\n",
    "exp5 = sym.Eq(x,x1 + vc*(t-t1))\n",
    "expt12 = sym.Eq(t2,t1 + dt_12)\n",
    "exph2 = sym.Eq(h2,exp4.args[1].subs({t:t2}))\n",
    "expx2 = sym.Eq(x2,exp5.args[1].subs({t:t2}))\n",
    "_t2 = expt12.args[1].subs(sub); sub[t2]=_t2\n",
    "_h2 = exph2.args[1].subs(sub); sub[h2]=_h2\n",
    "_x2 = expx2.args[1].subs(sub); sub[x2]=_x2\n",
    "display(exp4,exp5,expt12,exph2,expx2)\n",
    "_t2,_x2,_h2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "h = \\dot{h}⋅(t - t₂) + h₂",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle h = \\dot{h} \\left(t - t_{2}\\right) + h_{2}$"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "x = v_c⋅(t - t₂) + x₂",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle x = v_{c} \\left(t - t_{2}\\right) + x_{2}$"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "          -h₂ + h₃\nt₃ = t₂ + ────────\n          \\dot{h} ",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle t_{3} = t_{2} + \\frac{- h_{2} + h_{3}}{\\dot{h}}$"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "                                             2     \n               │\\dot{h}│            │\\dot{h}│      \nh₃ = - \\dot{h}⋅│───────│ - 0.5⋅da_z⋅│───────│  + h₄\n               │  da_z │            │  da_z │      ",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle h_{3} = - \\dot{h} \\left|{\\frac{\\dot{h}}{da_{z}}}\\right| - 0.5 da_{z} \\left|{\\frac{\\dot{h}}{da_{z}}}\\right|^{2} + h_{4}$"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "x₃ = v_c⋅(-t₂ + t₃) + x₂",
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAM8AAAAVCAYAAADhJLoWAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAGC0lEQVR4Ae2b7XUUNxSGFx8KcEIHTgcGKsB0wEcFmA7C4Zf9j0M6gFSQQAeBCiB0AKkgjjtwnkeRZmfGMzuj+dhdh7nnCGmkK817PyXNmtXV1dVqKXk6ODs7O1p0lqcz9fV/09vBaqEsDZyfn//MhOOsSQtz0sBR1F96vtH1EjwZ5sPwj2C/Q/0+Y9rCGjWA3j7YjHq88Xq55Xa6ULcGMPghXB+p73Zz53PE9f9k5kPa3/JX2M6MKXCyhnI+oL7cDup53rLsPP31+hrWN/3ZszlPmOGxZrbAYe0Tymk2suqEKXCqR/U5mCaSZfD7nbgET3/1PcFgb/uzZ3M+ZMaX7Fl5E9w9LWNoNM6oR/U5BssUsozRwxI8fbRnloNvth0hYvAd4U7QB9MOeabCqT6f7FCO0a++nVaIWSBt6ffpf0Y5ojyNPJ/g2duLcgn/T+D9yvMvEfeK9lfaL6iH4n/M/MkdGzw64nOKWVRdH9P3jlpdF/h53inNhFN9uovNuZs36g151PdoXy+Ch8Ves6iGXFH7OfZXikbU6dIZdajzuWyguJZOk0NfmKcDb6KXEatfxMRedr4feTYJDMV/j7mT33fAqwN9oBaz9xGdae9oJpwmtFw/mEo3k/h6CB6UYxTWL3Aa9EVEq/MV4/CbJR2/pBjF7lSv6O88s8MTAhT+ySji+RQX1AHrRyx3UTEOJWW8GDq5xzwxX9Mdcvnel3G+Opee0a/ed0FtOIf4g/pUvq0Susv19VYbpJ3nM4uWHS4oKfVR17O+gfSN/pDdqd2pPlJ+2Kom1i+7AEPaVTxHv1oPhZbOloKrNtTr0eTR6LC8V+Uqu3Vfesy8crCYgRP+8hpFhrSTOe5+fub1aNpKka8pqyuH6zQlsD67eytOls31B/0tJQRhNdIMsmT7ellfEU+wQQgeOsqGVAiVlHadJqHqznkHpnLwNc2ZrQ/8wbGpxa0T18/R3iXKx7jJsMR3D/7th/ni1Yn+aAB1yvg7SrpvmbTsU566zYrpjDUFx4p+Twt+Ds/WBXM24RziD60JqRCEBu+dVBbWq+uty9dbbRCCpwbWxaQiE0bFKUhy0joAjVIc68Lsln9Yw+yZ3tHCda27T1Z0kjukGTDgLK1icFcIHvH+TZH3kOdNDmVi0HnmoHsuyvtTgNgWj7h0nM+O7wG14gTrEH9Qnxe7lAvcyQ9bfR18rTa4raFg8AuPRwQNGCKddnknCZdxxioEj8c1L+JvaNezfYU3PcDXmEnS+MjaDF4xJO8zsH8rr0ufl9Xn1F7Wj2l77NoUPBq584gBzxCq3CMiXmW4pF3XqbozOVRkHPLSAXNacaa1wJXjD+48ZR9Ly8xWgy/b1zfZ4ACkRp/Fe0PKBpdJAvoca7wvMKbDPaD4JyUqbtekUxVODibb96kLZ6PtzqcDpkyvAZVhEzl/zAeHTWuLMTgRmNR/418ZMGaQmwgGHxGZO4Y6cYIxxx+Uo+moOgZj19zBvu7CdRsc0KcTmeFc+BQGs5tfdNxN/DLhEaLY1niuEGMGmvcjdy4NvDPi/eLwQigWsZ/EvjImPygYQIEYN8MXwZX6a7VGnks2Matj8fqr+7UdkD4d12PmXdpFYuN5m9SJUzARXx9/0N9a/WomwQb7epMNPLZpjMpRij6FahSMMbPjXxT/sC85XTKo5+LUR3P7BKaKLA0IxJ+FkTU93q0oGy/qDe/q7GJNdx2PRI3EeAgc6sAjBhgN+G0feRpxgiPbH5ijTKsdyKCfVvwDDK2+LkYJnkYbHPw33P9fFhLABaVsPLOI/b9T9p0MnGC8BBSZ+nzskKei+DR/rjoazV3S39AMXANHDOp/CGkjy2QEpiH+EHamkSAml6UJzyYbDPovCdGIT3mZX6sk7wP+JUI5oMLAvv0TlaHx/FYvHVLe0t/pVPB4fPNDw1bk5D3/RHxUa6L/1vpp9y3wGNS9/AFeE5dXgtbddvcSrRGAs9UGg4JnvfT31UKRBpq/u9wIw++jddCdCcgfiTuT1T7iL2NagqesjR5tjG7mPKGuf0buMfv7ZkFnfhTx/riVnXtubf8L9kwuZs8PiBsAAAAASUVORK5CYII=\n",
      "text/latex": "$\\displaystyle x_{3} = v_{c} \\left(- t_{2} + t_{3}\\right) + x_{2}$"
     },
     "metadata": {}
    },
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "(4.2, 3.38, 6.875)",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle \\left( 4.2, \\  3.38, \\  6.875\\right)$"
     },
     "metadata": {},
     "execution_count": 36
    }
   ],
   "source": [
    "# expressions for x and h during steady climb\n",
    "# For all t \\in [t2,t3], the following is true for h and x\n",
    "exp12 = sym.Eq(h,h2 + hdot*(t-t2))\n",
    "exp13 = sym.Eq(x,x2 + vc*(t-t2))\n",
    "dt_34 = abs(hdot/daz)\n",
    "exp9 = sym.Eq(h3,h4 - (hdot*(dt_34) + 0.5*daz*(dt_34)**2))\n",
    "exp11 = sym.Eq(t3,t2 + (h3-h2)/hdot)\n",
    "exp8 = sym.Eq(x3,x2 + vc*(t3-t2))\n",
    "\n",
    "display(exp12,exp13,exp11,exp9,exp8)\n",
    "_h3 = exp9.args[1].subs(sub); sub[h3]=_h3\n",
    "_t3 = exp11.args[1].subs(sub); sub[t3] = _t3\n",
    "_x3 = exp8.args[1].subs(sub); sub[x3] = _x3\n",
    "_t3,_x3,_h3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "          │\\dot{h}│\nt₄ = t₃ + │───────│\n          │  da_z │",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle t_{4} = t_{3} + \\left|{\\frac{\\dot{h}}{da_{z}}}\\right|$"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "                                        2     \nh = \\dot{h}⋅(t - t₃) + 0.5⋅da_z⋅(t - t₃)  + h₃",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle h = \\dot{h} \\left(t - t_{3}\\right) + 0.5 da_{z} \\left(t - t_{3}\\right)^{2} + h_{3}$"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "x = v_c⋅(t - t₃) + x₃",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle x = v_{c} \\left(t - t_{3}\\right) + x_{3}$"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "h₄ = h₅",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle h_{4} = h_{5}$"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "x₄ = v_c⋅(-t₃ + t₄) + x₃",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle x_{4} = v_{c} \\left(- t_{3} + t_{4}\\right) + x_{3}$"
     },
     "metadata": {}
    },
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "(6.7, 5.38, 10)",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle \\left( 6.7, \\  5.38, \\  10\\right)$"
     },
     "metadata": {},
     "execution_count": 37
    }
   ],
   "source": [
    "\n",
    "exp10 = sym.Eq(t4,t3 + dt_34) \n",
    "# For t \\in [t3,t4]\n",
    "exp14 = sym.Eq(h,(h3 + hdot*(t-t3) + 0.5*daz*(t-t3)**2))\n",
    "exp15 = sym.Eq(x,(x3 + vc*(t-t3)))\n",
    "exph4 = sym.Eq(h4,h5)\n",
    "expx4 = sym.Eq(x4,exp15.args[1].subs({t:t4}))\n",
    "display(exp10,exp14,exp15,exph4,expx4)\n",
    "_t4 = exp10.args[1].subs(sub); sub[t4] = _t4\n",
    "_x4 = expx4.args[1].subs(sub); sub[x4] = _x4\n",
    "_h4 = exph4.args[1].subs(sub)\n",
    "_t4,_x4,_h4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "          v₁ - v_c\nt₅ = t₄ + ────────\n             aₓ   ",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle t_{5} = t_{4} + \\frac{v_{1} - v_{c}}{a_{x}}$"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "                                         2\n          v_c⋅(v₁ - v_c)   0.5⋅(v₁ - v_c) \nx₅ = x₄ + ────────────── + ───────────────\n                aₓ                aₓ      ",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle x_{5} = x_{4} + \\frac{v_{c} \\left(v_{1} - v_{c}\\right)}{a_{x}} + \\frac{0.5 \\left(v_{1} - v_{c}\\right)^{2}}{a_{x}}$"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "h₅ = dh + h₀",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle h_{5} = dh + h_{0}$"
     },
     "metadata": {}
    },
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "(6.83333333333333, 5.5, 10)",
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      "text/latex": "$\\displaystyle \\left( 6.83333333333333, \\  5.5, \\  10\\right)$"
     },
     "metadata": {},
     "execution_count": 38
    }
   ],
   "source": [
    "# Time to reach v1 from vc with accelration ax: (ax is deceleration if vc > v1)\n",
    "dt_45 = (v1 - vc)/(ax)\n",
    "\n",
    "# Distance traveled from x4 to x5\n",
    "dx_45 = vc*dt_45 + 1/2*(ax)*dt_45**2\n",
    "\n",
    "expt5 = sym.Eq(t5,t4+dt_45)\n",
    "expx5 = sym.Eq(x5,x4 + dx_45)\n",
    "exph5 = sym.Eq(h5,h0+dh)\n",
    "display(expt5,expx5,exph5)\n",
    "_t5 = expt5.args[1].subs(sub); sub[t5] = _t5\n",
    "_x5 = expx5.args[1].subs(sub); sub[x5] = _x5\n",
    "_h5 = exph5.args[1].subs(sub)\n",
    "_t5,_x5,_h5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
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\n"
     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "# Plot points and vertical profile\n",
    "\n",
    "f1 = sym.lambdify(t,exp5.args[1].subs(sub))\n",
    "f2 = sym.lambdify(t,exp4.args[1].subs(sub))\n",
    "f3 = sym.lambdify(t,exp13.args[1].subs(sub))\n",
    "f4 = sym.lambdify(t,exp12.args[1].subs(sub))\n",
    "f5 = sym.lambdify(t,exp15.args[1].subs(sub))\n",
    "f6 = sym.lambdify(t,exp14.args[1].subs(sub))\n",
    "t0t1 = [i for i in np.arange(_t0,_t1+0.1,0.1)]\n",
    "t1t2 = [i for i in np.arange(_t1,_t2+0.1,0.1)]\n",
    "t2t3 = [i for i in np.arange(_t2,_t3+0.1,0.1)]\n",
    "t3t4 = [i for i in np.arange(_t3,_t4+0.1,0.1)]\n",
    "plt.scatter(0,0,marker='o')\n",
    "plt.scatter(_x1,0,marker='o')\n",
    "plt.scatter(_x2,_h2,marker='o')\n",
    "plt.scatter(_x3,_h3,marker='o')\n",
    "plt.scatter(_x4,_h4,marker='o')\n",
    "plt.scatter(_x5,_h5,marker='o')\n",
    "plt.annotate(\"$p_0$\",(0+0.1,0))\n",
    "plt.annotate(\"$p_1$\",(_x1+0.1,_h1))\n",
    "plt.annotate(\"$p_2$\",(_x2+0.1,_h2))\n",
    "plt.annotate(\"$p_3$\",(_x3+0.1,_h3))\n",
    "plt.annotate(\"$p_4$\",(_x4+0.1,_h4))\n",
    "plt.annotate(\"$p_5$\",(_x5+0.1,_h5))\n",
    "plt.plot([float(f1(i)) for i in t1t2],[float(f2(i)) for i in t1t2])\n",
    "plt.plot([float(f3(i)) for i in t2t3],[float(f4(i)) for i in t2t3])\n",
    "plt.plot([float(f5(i)) for i in t3t4],[float(f6(i)) for i in t3t4])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "'Success criteria:'"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "-x₀ + x₅ ≤ dx",
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIUAAAASCAYAAAB1uR3HAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAELUlEQVRoBe2Z7VHcMBCGDXMFZEIH0EEYKgA6CEkFhA5g+Mc/hnQAVBCgA6ACPjoIHUDSAXken2WUyxmkszF3THZmT2t9rN5drVayb+7x8bH4T+/PA3t7e5+w6hhehE943kq1cpDa8X+//jzAAn5mtl14E/l2kpmrccuU7vrzHB0zHxQYvYbBi5RHOYZPY19s+AauHfgCXuX5dxuclW9Uob5kmvmgwNIPFScbPW0dWbxtMJnez2B3d6tgiOxbR77L1fcegiLyweyILJTB7BHhUXHI89IroDeLZmUJMXQSFJWBpj5pBd6EveB8haUr+rgLpo76xl7Nd4AjXLADnjsJBvTob48es8w9bDB42dyHa0qxd77u3U7QuO8yaq5gb71rPAvSHaETppV6w44/DnHCJXyOvAR3cg9CjwF2A2vLDuw6BJ+PZooX7W0dFAAwQwQAiCWZEkNm+Ig82j7s9ca/b4BdX3RK2OCmO4UNhrtIufItdWaOklLtHVRKjV6Vp9IG48Kr0jVyDMbLjWDKOsqNUaXUhSAxzS3wbEZ5lujjLnNHjFLpaNrHvYeL45/5IwVZ2NFlinae8Irnsxe50d0YTfEkiqXS4QLqA3dt22yhHtfu5GmmUtJXYWOGpiR7DQojaTmMyi0ZH4IjDBVM4yJXzrinNMUVlB4zplODqZFoH7foBfVmJV9JS32NCsY0MCYLu/PAZkbfFgz6M3RkLSr9HbdF6ULuUv6k9KKZjZ9x0hf4gvGuY0mVbrGG4A31SfYOhmq6+QVM2Ml1hFYAC8oAWqeuhhmp1yCDwoWNM07o0kvJ3CnYxeL3g1HnZmNEh/5w85g1tmGDQ7/tV22IzxP9DCx5FI+BUtDemMFoa7S3VVBUoDzPTIMCKHczcry47oYyc1AavRqhQ2Ly2ZtyPC5u71wGiziSsccAGKtDHf/XDo375MjoM0t4UXfD3FDqS4Nl1E9Nar3cx+SRWQYKOsykyg9wkr3zdGxDOkd+YHKd5MS1IdTZFgO2j2S/mHzu/BIWTzBGzsUeVPhtweB14Y4rp4e2ViW6jmBfUU37l8hulEaiXV/fwm62kqgzsHy+HtYUK9SJN9neQTVw0kLHeKY6YcHknpUeBV4KfUUyWOqjxD4NZECEgGno0nl1NnZscYxcEs/7CNo5N6zp5rfyWYrfnNCj2OD0nrMA/4C9n5m9rfNZSrZ3rs9/SQFpBHt2+o5eHxXI/mnjzTzVEXQfEmMmvmgGHamlc8UYkQ3kX/A6ch0sqfqmtV/b4yPLLhxnIJjyxmUF0+AkpD65DzrFhjilhyOvDvA+QLz2HG2Pj0nwheOmDAKc7NFTf9fIVdjzDvUyGAdv+ZGOuheDgj5uBI+aHIq/B+WMa9W31+MjIMVBfnCR/HjlxSrnpu24N6FqYb1oitvzu6Cu8ZuM7bNIfwCXYfvW7PAY4AAAAABJRU5ErkJggg==\n",
      "text/latex": "$\\displaystyle - x_{0} + x_{5} \\leq dx$"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "x₅ = 5.5",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle x_{5} = 5.5$"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "dx = 8.6423",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle dx = 8.6423$"
     },
     "metadata": {}
    }
   ],
   "source": [
    "display(\"Success criteria:\")\n",
    "success = sym.LessThan(x5-x0,dx)\n",
    "display(success)\n",
    "lhs1 = sym.Eq(x5,expx5.args[1].subs(sub))\n",
    "rhs = sym.Eq(dx,dx.subs(sub))\n",
    "display(lhs1,rhs)"
   ]
  }
 ]
}